The Theory of the Riemann Zeta-function


Author: Late Savilian Professor of Geometry E C Titchmarsh,Titchmarsh, Edward Charles Titchmarsh,Edward Charles Titchmarsh,D. R. Heath-Brown
Publisher: Oxford University Press
ISBN: 9780198533696
Category: Architecture
Page: 412
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The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects of the theory, starting from first principles and probing the function's own challenging theory, with the famous and still unsolved "Riemann hypothesis" at its heart. The second edition has been revised to include descriptions of work done in the last forty years and is updated with many additional references; it will provide stimulating reading for postgraduates and workers in analytic number theory and classical analysis.

The Theory of Functions


Author: Edward Charles Titchmarsh
Publisher: Oxford University Press
ISBN: 0198533497
Category: Mathematics
Page: 454
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The Theory of Functions

The Riemann Zeta-Function

Theory and Applications
Author: Aleksandar Ivic
Publisher: Courier Corporation
ISBN: 0486140040
Category: Mathematics
Page: 562
View: 2636

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This text covers exponential integrals and sums, 4th power moment, zero-free region, mean value estimates over short intervals, higher power moments, omega results, zeros on the critical line, zero-density estimates, and more. 1985 edition.

The Riemann Zeta-Function


Author: Anatoly A. Karatsuba,S. M. Voronin
Publisher: Walter de Gruyter
ISBN: 3110886146
Category: Mathematics
Page: 408
View: 5869

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The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany

The Riemann Hypothesis and the Roots of the Riemann Zeta Function


Author: Samuel W. Gilbert
Publisher: Riemann hypothesis
ISBN: 9781439216385
Category: Mathematics
Page: 140
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The author demonstrates that the Dirichlet series representation of the Riemann zeta function converges geometrically at the roots in the critical strip. The Dirichlet series parts of the Riemann zeta function diverge everywhere in the critical strip. It has therefore been assumed for at least 150 years that the Dirichlet series representation of the zeta function is useless for characterization of the non-trivial roots. The author shows that this assumption is completely wrong. Reduced, or simplified, asymptotic expansions for the terms of the zeta function series parts are equated algebraically with reduced asymptotic expansions for the terms of the zeta function series parts with reflected argument, constraining the real parts of the roots of both functions to the critical line. Hence, the Riemann hypothesis is correct. Formulae are derived and solved numerically, yielding highly accurate values of the imaginary parts of the roots of the zeta function.

Sieve Methods


Author: Heine Halberstam,Hans Egon Richert
Publisher: Courier Corporation
ISBN: 0486320804
Category: Mathematics
Page: 384
View: 393

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This text by a noted pair of experts is regarded as the definitive work on sieve methods. It formulates the general sieve problem, explores the theoretical background, and illustrates significant applications. 1974 edition.

Lectures on the Riemann Zeta Function


Author: H. Iwaniec
Publisher: American Mathematical Society
ISBN: 1470418517
Category: Mathematics
Page: 119
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The Riemann zeta function was introduced by L. Euler (1737) in connection with questions about the distribution of prime numbers. Later, B. Riemann (1859) derived deeper results about the prime numbers by considering the zeta function in the complex variable. The famous Riemann Hypothesis, asserting that all of the non-trivial zeros of zeta are on a critical line in the complex plane, is one of the most important unsolved problems in modern mathematics. The present book consists of two parts. The first part covers classical material about the zeros of the Riemann zeta function with applications to the distribution of prime numbers, including those made by Riemann himself, F. Carlson, and Hardy-Littlewood. The second part gives a complete presentation of Levinson's method for zeros on the critical line, which allows one to prove, in particular, that more than one-third of non-trivial zeros of zeta are on the critical line. This approach and some results concerning integrals of Dirichlet polynomials are new. There are also technical lemmas which can be useful in a broader context.

The Theory of Hardy's Z-Function


Author: A. Ivić
Publisher: Cambridge University Press
ISBN: 1107028833
Category: Mathematics
Page: 245
View: 5962

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"This book is an outgrowth of a mini-course held at the Arctic Number Theory School, University of Helsinki, May 18-25, 2011. The central topic is Hardy's function, of great importance in the theory of the Riemann zeta-function. It is named after GodfreyHarold Hardy FRS (1877-1947), who was a prominent English mathematician, well-known for his achievements in number theory and mathematical analysis"--

Exploring the Riemann Zeta Function

190 years from Riemann's Birth
Author: Hugh Montgomery,Ashkan Nikeghbali,Michael Th. Rassias
Publisher: Springer
ISBN: 3319599690
Category: Mathematics
Page: 298
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Exploring the Riemann Zeta Function: 190 years from Riemann's Birth presents a collection of chapters contributed by eminent experts devoted to the Riemann Zeta Function, its generalizations, and their various applications to several scientific disciplines, including Analytic Number Theory, Harmonic Analysis, Complex Analysis, Probability Theory, and related subjects. The book focuses on both old and new results towards the solution of long-standing problems as well as it features some key historical remarks. The purpose of this volume is to present in a unified way broad and deep areas of research in a self-contained manner. It will be particularly useful for graduate courses and seminars as well as it will make an excellent reference tool for graduate students and researchers in Mathematics, Mathematical Physics, Engineering and Cryptography.

Mathematics for the General Reader


Author: E.C. Titchmarsh
Publisher: Courier Dover Publications
ISBN: 0486813924
Category: Mathematics
Page: 192
View: 5068

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Numerous helpful examples clarify this accessible treatment of algebra, fractions, geometry, irrational numbers, logarithms, infinite series, complex numbers, quadratic equations, trigonometry, functions, and integral and differential calculus.

Fractal Geometry and Number Theory

Complex Dimensions of Fractal Strings and Zeros of Zeta Functions
Author: Michel Lapidus,Machiel van Frankenhuysen
Publisher: Springer Science & Business Media
ISBN: 1461253144
Category: Mathematics
Page: 268
View: 2927

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An Introduction to the Theory of Numbers


Author: Godfrey Harold Hardy,E. M. Wright,Roger Heath-Brown,Joseph Silverman
Publisher: Oxford University Press
ISBN: 9780199219865
Category: Mathematics
Page: 621
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An Introduction to the Theory of Numbers by G.H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. This Sixth Edition has been extensively revised and updated to guide today's students through the key milestones and developments in number theory. Updates include a chapter on one of the mostimportant developments in number theory -- modular elliptic curves and their role in the proof of Fermat's Last Theorem -- a foreword by A. Wiles and comprehensively updated end-of-chapter notes detailing the key developments in number theory. Suggestions for further reading are also included for the more avid reader and the clarityof exposition is retained throughout making this textbook highly accessible to undergraduates in mathematics from the first year upwards.

The Riemann Hypothesis

A Resource for the Afficionado and Virtuoso Alike
Author: Peter Borwein
Publisher: Springer Science & Business Media
ISBN: 0387721258
Category: Mathematics
Page: 533
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This book presents the Riemann Hypothesis, connected problems, and a taste of the body of theory developed towards its solution. It is targeted at the educated non-expert. Almost all the material is accessible to any senior mathematics student, and much is accessible to anyone with some university mathematics. The appendices include a selection of original papers. This collection is not very large and encompasses only the most important milestones in the evolution of theory connected to the Riemann Hypothesis. The appendices also include some authoritative expository papers. These are the "expert witnesses” whose insight into this field is both invaluable and irreplaceable.

Limit Theorems for the Riemann Zeta-Function


Author: Antanas Laurincikas
Publisher: Springer Science & Business Media
ISBN: 9401720916
Category: Mathematics
Page: 306
View: 4734

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The subject of this book is probabilistic number theory. In a wide sense probabilistic number theory is part of the analytic number theory, where the methods and ideas of probability theory are used to study the distribution of values of arithmetic objects. This is usually complicated, as it is difficult to say anything about their concrete values. This is why the following problem is usually investigated: given some set, how often do values of an arithmetic object get into this set? It turns out that this frequency follows strict mathematical laws. Here we discover an analogy with quantum mechanics where it is impossible to describe the chaotic behaviour of one particle, but that large numbers of particles obey statistical laws. The objects of investigation of this book are Dirichlet series, and, as the title shows, the main attention is devoted to the Riemann zeta-function. In studying the distribution of values of Dirichlet series the weak convergence of probability measures on different spaces (one of the principle asymptotic probability theory methods) is used. The application of this method was launched by H. Bohr in the third decade of this century and it was implemented in his works together with B. Jessen. Further development of this idea was made in the papers of B. Jessen and A. Wintner, V. Borchsenius and B.

Riemann's Zeta Function


Author: Harold M. Edwards
Publisher: Courier Corporation
ISBN: 9780486417400
Category: Mathematics
Page: 315
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Superb high-level study of one of the most influential classics in mathematics examines landmark 1859 publication entitled “On the Number of Primes Less Than a Given Magnitude,” and traces developments in theory inspired by it. Topics include Riemann's main formula, the prime number theorem, the Riemann-Siegel formula, large-scale computations, Fourier analysis, and other related topics. English translation of Riemann's original document appears in the Appendix.

The Gamma Function


Author: Emil Artin
Publisher: Courier Dover Publications
ISBN: 0486789780
Category: Mathematics
Page: 48
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"This brief monograph on the gamma function by a major 20th century mathematician was designed to bridge a gap in the literature of mathematics between incomplete and over-complicated treatments. Topics include functions, the Euler integrals and the Gauss formula, large values of X and the multiplication formula, the connection with sin X applications to definite integrals, and other subjects. "--

Topics in Analytic Number Theory


Author: Hans Rademacher
Publisher: Springer Science & Business Media
ISBN: 3642806155
Category: Mathematics
Page: 322
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At the time of Professor Rademacher's death early in 1969, there was available a complete manuscript of the present work. The editors had only to supply a few bibliographical references and to correct a few misprints and errors. No substantive changes were made in the manu script except in one or two places where references to additional material appeared; since this material was not found in Rademacher's papers, these references were deleted. The editors are grateful to Springer-Verlag for their helpfulness and courtesy. Rademacher started work on the present volume no later than 1944; he was still working on it at the inception of his final illness. It represents the parts of analytic number theory that were of greatest interest to him. The editors, his students, offer this work as homage to the memory of a great man to whom they, in common with all number theorists, owe a deep and lasting debt. E. Grosswald Temple University, Philadelphia, PA 19122, U.S.A. J. Lehner University of Pittsburgh, Pittsburgh, PA 15213 and National Bureau of Standards, Washington, DC 20234, U.S.A. M. Newman National Bureau of Standards, Washington, DC 20234, U.S.A. Contents I. Analytic tools Chapter 1. Bernoulli polynomials and Bernoulli numbers ....... . 1 1. The binomial coefficients ..................................... . 1 2. The Bernoulli polynomials .................................... . 4 3. Zeros of the Bernoulli polynomials ............................. . 7 4. The Bernoulli numbers ....................................... . 9 5. The von Staudt-Clausen theorem .............................. . 10 6. A multiplication formula for the Bernoulli polynomials ........... .

Fractal Geometry, Complex Dimensions and Zeta Functions

Geometry and Spectra of Fractal Strings
Author: Michel Lapidus,Machiel van Frankenhuijsen
Publisher: Springer Science & Business Media
ISBN: 1461421756
Category: Mathematics
Page: 570
View: 5461

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Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. Throughout Geometry, Complex Dimensions and Zeta Functions, Second Edition, new results are examined and a new definition of fractality as the presence of nonreal complex dimensions with positive real parts is presented. The new final chapter discusses several new topics and results obtained since the publication of the first edition.